Explicit open images for elliptic curves over $\mathbb{Q}$

Author name not available (Why is that?)

Abstract: For a non-CM elliptic curve

E

defined over

m a t h b b Q

, the Galois action on its torsion points gives rise to a Galois representation

h o_{E} : G a l (o v e r l i n e m a t h b b Q / m a t h b b Q) o G L_{2} (w i d e h a t m a t h b b Z)

that is unique up to isomorphism. A renowned theorem of Serre says that the image of

h o_{E}

is an open, and hence finite index, subgroup of

G L_{2} (w i d e h a t m a t h b b Z)

. We describe an algorithm that computes the image of

h o_{E}

up to conjugacy in

G L_{2} (w i d e h a t m a t h b b Z)

; this algorithm is practical and has been implemented. Up to a positive answer to a uniformity question of Serre and finding all the rational points on a finite number of explicit modular curves of genus at least

2

, we give a complete classification of the groups

h o_{E} (G a l (o v e r l i n e m a t h b b Q / m a t h b b Q)) c a p S L_{2} (w i d e h a t m a t h b b Z)

and the indices

[G L_{2} (w i d e h a t m a t h b b Z) : h o_{E} (G a l (o v e r l i n e m a t h b b Q / m a t h b b Q))]

for non-CM elliptic curves

E / m a t h b b Q

. Much of the paper is dedicated to the efficient computation of modular curves via modular forms expressed in terms of Eisenstein series.

Has companion code repository: https://github.com/davidzywina/openimage

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